Determine the exact value: cot(5pi/6)

cot(5Π/6)

= cot(Π - Π/6)

= - cot(Π/6)

= -√3

cot((5π)/(6))

Take the cotangent of (5π)/(6) to get -(3)/(√(3)).
-(3)/(√(3))

To rationalize the denominator of a fraction, rewrite the fraction so that the new fraction has the same value as the original and has a rational denominator. The factor to multiply by should be an expression that will eliminate the radical in the denominator. In this case, the expression that will eliminate the radical in the denominator is (√(3))/(√(3)).
-(3)/(√(3))*(√(3))/(√(3))

To eliminate the radical from the denominator, multiply √(3) by √(3) to get 3.
(-3*√(3))/(3)

Multiply -3 by √(3) to get -3√(3).
(-3√(3))/(3)

Move the minus sign from the numerator to the front of the expression.
-(3√(3))/(3)

Reduce the expression -(3√(3))/(3) by removing a factor of 3 from the numerator and denominator.
-√3